On the dynamical Yang-Baxter equation

Abstract

This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is an important equation arising in quantum field theory and statistical mechanics, is best understood within the framework of Hopf algebras, or quantum groups. On the other hand, in Felder's talk, it is explained that another important equation of mathematical physics, the star-triangle relation, may (and should) be viewed as a generalization of QYBE, in which solutions depend on additional ``dynamical'' parameters. It is also explained there that to a solution of the quantum dynamical Yang-Baxter equation one may associate a kind of quantum group. These ideas gave rise to a vibrant new branch of ``quantum algebra'', which may be called the theory of dynamical quantum groups. My goal in this talk is to give a bird's eye review of some aspects of this theory and its applications.

An updated version of the ICM-2002 talk by the author

Cited by in corpus (11)

• Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection
• Chiral zero modes of the SU(n) Wess-Zumino-Novikov-Witten model
• On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras
• An infinite product formula for $U_q(sl(2))$ dynamical coboundary element
• The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space
• Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras
• Classification of Non-Affine Non-Hecke Dynamical R-Matrices
• Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices
• Constructing dynamical twists over a non-abelian base
• Super Solutions of the Dynamical Yang-Baxter Equation
• Dynamical twists in Hopf algebras